express x² - 8x + 27 in the form (x+a)² +b?

Question

plzz help

im confused

tanx

how would you deduce the least possible value of this expression?

Answer 1

This is a process called 'completing the square'

First, expand out (x+a)² +b
= x² + 2ax + a² + b

Then compare the coefficients of this quadratic with those in the quadratic you are given (x² - 8x + 27)

We see that 2a = -8 and a² + b = 27
These equations are enough for you to pin down both a and b.

Answer 2

The process that is required here is called completing the square.

Generally, any quadratic expression of the form:

ax^2 + bx + c [1]

can be expressed in the form (x + b/2)^2 - (b/2)^2. [2]

In your example you have:

x^2 - 8x + 27 so comparing this with [1] you have a= 1, b= -8 and c = 27. Don't get confused or make a mistake at this point because these values for a and b are NOT the same as the a and b in the RHS (x + a)^2 + b.

So we have b = -8 from [1] and we now put this value for the b in expression [2] and get:

(x + (-8/2)^2 - (-8/2)^2 + 27

= (x - 4)^2 - (64/4) + 27

= (x - 4)^2 - 16 + 27

= (x -4)^2 + 11 which is of the form (x+a)^2 + b where

a = (-4) and b = 11

Answer 3

Simply mutiply out, the coeff of x^2, x and constants must be identical..
x^2+2ax+a^2+b is identical to x^2-8x+27..So..
2a=-8> a=-4 and a^2+b=27>b=11
> (x-4)^2+11 min value by differentiating x^2-8x+27 set y'=0
>x=4 or by inspection of (x-4)^2+11..min value =11..sketch parabola and you'll see! function has a min TV at (4,11)

Answer 4

to do these kinds of equations you look at the middle term (-8x)
divide this by two to get -4. You can do this because when doing (x+a)(x+a) - the x term added up will be ax + ax = 2ax. So -4x + -4x = 8x. ok.

So you have (x-4)^2 or (x + (-4))^2. but that gives you + 16 at the end.
you need 11 more to make this equal to your original.

So it would (x-4)^2 + 11. if you do this out, you should get back to your original x^2 - 8x + 27

hope you understand these completing the square problems
:)

p.s. SV's answer is very good and complete without actually giving you the answer. Take a long careful read of it.

Answer 5

(x ² - 8 x + 16) - 16 + 27
( x - 4 ) ² + 11

Answer 6

x^2 - 8x + 27
= x^2 - 8x + 16 + 11
(as x^2 - 8x + 16 = (x-4)^2 and 16+11 = 27)
= (x - 4)^2 + 11

a = -4, b = 11

Answer 7

Determine what number, when added to itself will give you -8.

This is -4

(x - 4)² = x² - 8x + 16

So, you need another 11 to complete your original equation:

(x - 4)² + 11

Answer 8

(x-4)² +11

Answer 9

x^2-8x+27
(x-4)^2+11

Answer 10

(x+(-4))^2+11